Microelectronics
Journal
Microelectronics Journal 32 (2001) 931±941
www.elsevier.com/locate/mejo
Optimal wavelet denoising for phonocardiograms
Sheila R. Messer a, John Agzarian b, Derek Abbott a,*
a
Department of Electrical and Electronic Engineering, Centre for Biomedical Engineering (CBME), Adelaide University, Adelaide, SA 5005, Australia
b
Hampstead Medical Clinic, Hampstead Gardens, SA 5086, Australia
Revised 15 June 2001; accepted 21 August 2001
Abstract
Phonocardiograms (PCGs), recordings of heart sounds, have many advantages over traditional auscultation in that they may be replayed
and analysed for spectral and frequency information. PCG is not a widely used diagnostic tool as it could be. One of the major problems with
PCG is noise corruption. Many sources of noise may pollute a PCG including foetal breath sounds if the subject is pregnant, lung and breath
sounds, environmental noise and noise from contact between the recording device and the skin. An electronic stethoscope is used to record
heart sounds and the problem of extracting noise from the signal is addressed via the use of wavelets and averaging. Using the discrete
wavelet transform, the signal is decomposed. Due to the ef®cient decomposition of heart signals, their wavelet coef®cients tend to be much
larger than those due to noise. Thus, coef®cients below a certain level are regarded as noise and are thresholded out. The signal can then be
reconstructed without signi®cant loss of information in the signal content. The questions that this study attempts to answer are which wavelet
families, levels of decomposition, and thresholding techniques best remove the noise in a PCG. The use of averaging in combination with
wavelet denoising is also addressed. Possible applications of the Hilbert transform to heart sound analysis are discussed. q 2001 Published
by Elsevier Science Ltd.
Keywords: Wavelets; Denoising; Phonocardiogram; Heart sound analysis; Heartbeat analysis; Hilbert transform
1. Introduction
The stethoscope and human ear have their limitations in
diagnosing heart defects and conditions. Modern technology
has developed new tools, which are capable of revealing
information that traditional tools such as the stethoscope
alone cannot. For example, digital stethoscopes have been
developed, which have the capacity to record and replay the
heartbeat sound recordings, otherwise known as phonocardiograms (PCGs). The PCG is a particularly useful diagnostic tool because the graphic recordings show timings
and relative intensities of heartbeat sounds and may reveal
information that the human ear cannot [1±3]. With the aid
of computers, the PCG data may be stored, managed, and
manipulated for frequency and spectral content.
Electrocardiograms (ECGs), which reveal the electrical
activity of the heart, and echocardiography, which uses
ultrasound waves to create an image of the heart, are other
methods that are used to assess the condition the heart. All
of the mentioned techniques are non-invasive, but each
method presents a separate set of issues and challenges.
* Corresponding author. Tel.: 161-8-8303-5748; fax: 161-8-8303-4360.
E-mail addresses: smesser@eleceng.adelaide.edu.au (S.R. Messer),
dabbott@eleceng.adelaide.edu.au (D. Abbott).
0026-2692/01/$ - see front matter q 2001 Published by Elsevier Science Ltd.
PII: S 0026-269 2(01)00095-7
With the advent of echocardiography, auscultation and
phonocardiography have become less important. However,
with the aid of computers and digital signal processing techniques, PCGs may reveal important information [1].
The ECG is not normally used unless a problem has been
previously detected by auscultation (listening to the heart
sounds), because the time required to set up an ECG recording is longer and hence not used as a standard test. However,
PCGs are easily obtained by placing the stethoscope against
the skin. The current problem with many PCG systems is
noise from sources such as breath sounds, contact of the
stethoscope with the skin, foetal heart sounds if the subject
is pregnant, and ambient noise that may corrupt the heart
sound signals.
The PCG would be a much more useful diagnostic tool if
unwanted noise was removed revealing the heartbeat sound
clearly. The current study examines methods of removing
the noise from the PCG using namely wavelet analysis and
averaging. Currently, there is no way of knowing a priori
what the particular noise component is, or of determining
the noise component once the measurement has been
recorded. The previously mentioned sources of noise are
by no means a complete list. In every case and situation,
the noise will be different.
Although what produces each of the four heart sounds
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S.R. Messer et al. / Microelectronics Journal 32 (2001) 931±941
study examines the use of averaging in combination with
wavelet analysis to remove noise from a PCG.
2. Equipment
An electronic stethoscope (the Escope from Cardionics),
using an electret microphone outputs the heart sound as an
analogue signal. This analogue signal is converted to a digital signal (sampled at 2500 samples/s with 12-bit resolution)
and stored on the computer for further use. The electrical
activity of the heart is also simultaneously recorded to
serve as a reference signal. matlab software is then used
to analyse the signal and perform the signal processing. For
more detail on the equipment and recording procedure, refer
to our previous study [6].
Fig. 1. This ®gure shows the principle of denoising a heartbeat. (a) Is the
characteristic heartbeat signal (b) the heartbeat with 1 dB of additive white
noise (c) the heartbeat with noise removed using wavelet analysis.
heard through a stethoscope is not exactly known, they are
likely to be produced by a number of sources including the
opening and closing of valves, vibrations of the cardiac
structure, and acceleration and deceleration of blood [1±
4]. The ®rst and second heart sounds (`lub-dub') are the
two that are generally heard by the human ear, and are
usually the most visible on the PCG. They may be seen in
Fig. 1(a) with the ®rst heart sound, S1, occurring at about
0.1 s and the second heart sound, S2, happening at about
0.4 s.
An electronic stethoscope is used to record heart sounds.
Various digital signal processing tools namely wavelets are
employed to remove noise from signal. The remainder of
this article will introduce the reader to basic wavelet theory
in relation to heart sounds and the methods of denoising
heart sounds, which were investigated.
Wavelets may be used to denoise the PCG as shown in
Fig. 1. The signal is decomposed by a discrete wavelet
transform. Because of the ef®cient decomposition of heart
signals, their wavelet coef®cients tend to be much larger
than those due to noise. Thus, coef®cients below a certain
level are regarded as noise and thresholded out. The signal is
then reconstructed without signi®cant loss of information.
The questions that this study attempts to answer are,
which wavelet families, levels of decomposition, and
thresholding techniques best remove the noise in a PCG.
This study is an extension of the work described by Maple
et al. [5].
Averaging may also be used to produce a characteristic
heartbeat [4]. Although heartbeats are considered nonstationary signals, they are periodic in the sense that heartbeats regularly repeat. Over short periods of time, heartbeats
have the same statistical properties. Thus, the signal may be
considered quasi-stationary over a short period of time. This
3. Wavelet theory
Wavelet theory dates back to the work of Joseph Fourier,
but most of the advances in the ®eld have been made since
the 1980's. This section gives a review of basic theory
needed to understand wavelet denoising.
3.1. Fourier analysis
In 1822, Joseph Fourier discovered that any periodic
function could be represented as an in®nite sum of periodic
complex exponential functions [7]. The inclusive property
of only periodic functions was later extended to any discrete
time function. The Fourier transform (FT) converts a signal
expressed in the time domain to a signal expressed in the
frequency domain. The FT representation of a signal may
be seen in Fig. 2(a). The FT is widely used and usually
implemented in the form of the fast FT algorithm. The
mathematical de®nition of the FT is given below
Z2 1
X f
x te2j2pft dt
1
11
The time domain signal x t is multiplied by a complex
exponential at a frequency f and integrated over all time. In
other words, any discrete time signal may be represented by
a sum of sines and cosines, which are shifted and are multiplied by a coef®cient that changes their amplitude. X f are
the Fourier coef®cients which are large when a signal
contains a frequency component around the frequency f.
Fig. 2. Comparison of a signal represented in different domains with
(a) corresponding to the Fourier transform representation, (b) representing
the short time Fourier transform, and (c) the wavelet transform.
S.R. Messer et al. / Microelectronics Journal 32 (2001) 931±941
933
Fig. 3. Examples of wavelets used in this study.
The peaks in a plot of the FT of a signal correspond to
dominant frequency components of the signal.
The fast Fourier transform (FFT) is widely used, perhaps
even too widely used. Yves Meyer states [8], ªBecause the
FFT is very effective, people have used it in problems where
it is not useful-the way Americans use cars to go half a
block.º
Fourier analysis is simply not effective when used on nonstationary signals because it does not provide frequency
content information localized in time. Most real world
signals exhibit non-stationary characteristics (such as heart
sound signals). Thus, Fourier analysis is not adequate.
3.2. Short time Fourier transform
The problem with Fourier analysis is the fact that it does
not matter when frequency components appear in a signal
because the signal is integrated over all time in Eq. (1).
Thus, the frequency content of the signal is known but its
location in time is not known.
In an effort to answer this problem, the STFT was developed in 1946 by Denis Gabor [9]. The STFT analyses a
small section of the signal at a time, which is known as
windowing. The STFT is a compromise between the time
and frequency representation of a signal providing information on the frequency content, when it occurs. The trade-off
is between rather imprecise time and frequency resolution,
which is determined by the window size. The STFT representation of a signal may be seen in Fig. 2(b). The mathematical representation of STFT is
Z
STFTxw t 0 ; x x twp t 2 t 0 e2j2pft dt
2
t
where x t is the signal and w t is the windowing function
which is translated by a certain amount denoted as t 0 . The
windowing process translates the complex conjugate of the
window function along the length of the signal while multiplying the signal and windowing function at different points
in time. The function of the exponential component in Eq.
(2) is to convert the product of multiplication of the signal
and windowing function from the time domain to the
frequency domain.
The problem with the STFT is a compromise in resolution. The smaller the window used, the better quickly changing components are picked up, but slowly changing details
are not detected very well. If a larger window is used, lower
frequencies may be detected, but localization in time
becomes worse.
3.3. The wavelet transform
The Wavelet transform (WT) was developed as a method
to obtain simultaneous, high resolution time and frequency
information about a signal. The term `wavelet' was ®rst
mentioned in 1909 in a thesis by Alfred Haar [9], although
the progress in the ®eld of wavelets has been relatively slow
until the 1980's when scientists and engineers from different
®elds realized they were working on the same concept and
began collaborating [8].
The WT presents an improvement over the STFT because
it obtains good time and frequency resolution simultaneously by using a variable sized window region (the wavelet) instead of a constant window size. Because the wavelet
may be dilated or compressed as is seen in Fig. 2(c), different features of the signal are extracted. While a narrow
wavelet extracts high frequency components, a stretched
wavelet picks up on the lower frequency components of
the signal.
A wavelet is a signal of limited duration that has an
average value of zero. Examples of wavelets used in this
study may be seen in Fig. 3.
The mathematical description of the Continuous Wavelet
Transform (CWT) is described by
1 Z
c
p t2t
p
CWTC
x
t
t
;
s
C
t
;
x
c
dt
3
x
x
s
usu
The scale, s, of the wavelet may conceptually be considered the inverse of frequency. As seen in Fig. 2(c), the
wavelet is compressed if the scale is low and dilated if the
scale is high. Because the WT is computed in terms of scale
instead of frequency, plots of the WT of a signal are
displayed as time versus scale.
The process of computing the CWT is very similar to that
of the STFT. The wavelet is compared to a section at the
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S.R. Messer et al. / Microelectronics Journal 32 (2001) 931±941
Fig. 4. This ®gure shows a complex PCG trace ®rst with additive white noise and secondly without noise. The x-axis represents time, the y-axis represents the
PCG signal and the z-axis represents the HT of the PCG.
beginning of a signal. A number is calculated showing the
degree of correlation between the wavelet and signal
section. The wavelet is shifted right and the process is
repeated until the whole signal is covered. The wavelet is
scaled and the previous process is repeated for all scales.
The CWT reveals more detail about a signal, but because
all scales are used to compute the WT, the computation time
required can be enormous. Therefore, the discrete wavelet
transform (DWT) is normally used. The DWT calculates
the wavelet coef®cients at discrete intervals of time and
scale, instead of at all scales. The DWT requires much
less computation time than the CWT without much loss
in detail. With the DWT, a fast algorithm is possible
which possesses the same accuracy as other methods.
The algorithm makes use of the fact that if scales and
positions are chosen based on powers of two (dyadic
scales and positions) the analysis is very ef®cient. Because
the algorithm possesses the same accuracy as other methods, this method is often used and is used in the current
study.
4. Averaging theory
Averaging is known to reduce white noise because it is
randomly distributed throughout the signal and may also be
used to produce a `characteristic heartbeat' which is an
averaged heartbeat from a series of beats [4]. Over short
periods of time, heartbeats have the same statistical properties. Thus, the signal may be considered quasi-stationary
over a short period of time [4].
According to basic probability theory [10], the intensity
of
pa random signal averaging of n cycles is attenuated by
n: Thus, if 20 cycles were averaged, random signalspin
the
heartbeat series would be attenuated by a factor of 20 <
4:5 or if 50 cycles
p were averaged, the attenuation factor
would be about 50 < 7:
An important factor to consider in the use of averaging
to denoise heartbeats is the type of signal sought. The
mechanical activity of the heart can be classi®ed into two
categories: `deterministic' and `nondeterministic' [10]. In
our case, any process that repeats itself exactly for each
beat may be considered deterministic. For non-deterministic
events (e.g. such as heart murmurs), averaging the signal
will tend to reduce our ability to discriminate these from
deterministic characteristics in the heart sound signal Ð
this is where wavelet denoising can offer an advantage
over simple averaging.
The algorithm for averaging the PCG signal uses the
ECG as a gating signal because they are both recorded
simultaneously. The QRS complex of the ECG signals
the beginning of the cycle and is used to separate each
heartbeat. A description of the complete algorithm is given
by Tinati [4].
S.R. Messer et al. / Microelectronics Journal 32 (2001) 931±941
5. Hilbert transform theory
6. Denoising of heartbeats
The Hilbert transform (HT) may be used to calculate
the instantaneous attributes of a signal. The mathematical
de®nition of the Hilbert transform is [11]
6.1. Signal decomposition and reconstruction
y t p21
Z 2 1 x t
dt
t2t
1
4
The Hilbert transform (HT) can be considered a convolution between the signal and 1=pt: The HT can be realized
by an ideal ®lter whose amplitude response is unity and
whose phase response is a constant ninety degree lag. The
HT is called the quadrature ®lter because it shifts the phase
of the spectral components by p=2t:
Fig. 4 borrows the concept of a complex trace from seismic data analysis [12]. The signal and its HT are projected
on their prospective axes with the complex trace being a
vector sum of the two. This view reveals many features of
the signal. The length of the complex trace vector is the
instantaneous amplitude. The orientation angle (usually
measured relative to the positive axis of the plane where
the real signal is projected) is the instantaneous phase.
The time rate of change of the phase angle is the instantaneous frequency which may be seen in Fig. 5. The instantaneous frequency is calculated through the analytic method
using the HT. The instantaneous frequency is mathematically de®ned in Eq. (5) where s is the signal and H s is the
HT of the signal [13].
Q
1 d
Hs t
arctan
2p dt
s t
5
The instantaneous frequency of a signal may be used to
demonstrate the effectiveness of denoising [14].
935
The matlab wavelet toolbox provides built in routines
for using the DWT to decompose a signal into wavelet
coef®cients and then to reconstruct the signal using the
inverse discrete wavelet transform (IDWT). Many wavelet
families are available in the toolbox. However, in the
current study only orthogonal wavelets are examined
because they allow for perfect reconstruction of a signal.
The process used by matlab in decomposing and reconstructing a signal applies a series of highpass and lowpass
®lters in succession. This procedure was developed by
Mallat in 1988, and is a well known signal processing tool
called the two-channel subband coder [15]. The signal is
passed through highpass and lowpass ®lters and downsampled to keep the original number of datapoints. This
procedure results in details, which are low scale, high
frequency elements of the signal, and approximations,
which are high scale, low frequency elements of the signal.
This decomposition can be performed for many levels with
the decomposition process being iterated for successive
approximations [9,16].
6.2. White noise removal by thresholding
A classic problem with most noise removal methods is
actually knowing the noise content from a time series and
how to classify the signal. With wavelet denoising, it is not
necessary to know which part of the signal is white noise.
David Donoho and Iain Johnstone proved mathematically
that if a special kind of orthogonal basis existed, then it
would do the best job at removing white noise from a signal.
Then, in 1990, Donoho conversed with Domonique Picard
and Gerard Kerkyachaian and realized that they had found
the answer to the question of white noise removal using
wavelets [8]. The WT is applied to the signal and all coef®cients below a certain size are discarded [8]. This technique
makes use of the fact that some of the decomposed wavelet
coef®cients correspond to signal averages and others are
associated with details on the original signal. If the smaller
details are eliminated from the signal decomposition, the
original signal can be extracted from the remaining coef®cients and the main signal characteristics will remain intact
because an orthogonal wavelet transform compresses the
`energy' of the signal into a few large components and the
white noise is very disordered and hence it is scattered
throughout the transform in small coef®cients [8,17].
6.3. Optimal parameter selection for wavelet denoising of
PCGs
Fig. 5. This ®gure shows (a) a characteristic heartbeat as a reference (b) the
instantaneous frequency of a heartbeat. It is interesting to note that at the
®rst and second heart sound the dominant frequency appears to be about
10 Hz. There are artifacts present outside the principal areas of interest (S1
and S2) due to computational rounding errors.
When using wavelets to denoise PCGs, there are many
factors that must be considered. Examples of these choices
are which wavelet, level of decomposition, and thresholding
methods to use.
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S.R. Messer et al. / Microelectronics Journal 32 (2001) 931±941
6.3.1. Choosing the wavelets to use
In general, the more a wavelet resembles the signal, the
better it denoises the signal. However, because it is very
dif®cult and time-consuming to actually create a useful
wavelet, the wavelets which will be considered for use
will be previously de®ned.
matlab provides several families of wavelets including
the Morlet, Mexican hat, Meyer, Haar, Daubechies,
Symlets, Coi¯ets and Spline biorthogonal wavelets and
provides further documentation about these wavelet
families [9]. In order to obtain perfect reconstruction results,
only orthogonal wavelets will be considered. The orthogonal wavelet transform 1 has certain bene®ts. It is very
concise, allows for perfect reconstruction of the original
signal and is not very dif®cult to calculate [9]. The ease of
calculation exists because each wavelet coef®cient is calculated with only one scalar product of the signal and the
wavelet [8]. Being computationally inexpensive, orthogonal
wavelets allow a transform to be computed which contains
the same number of points as the original signal [8]. Wavelets that have the properties of orthogonality and a scaling
function f allow the use of the fast algorithm. The wavelets
which satisfy these criteria are the Haar, Daubechies,
Coi¯ets, and Symlets.
In matlab, the Daubechies family of wavelets consists of
45 wavelets. The Haar wavelet is the ®rst and most simple
wavelet in this family. The Daubechies family of wavelets is
not explicitly mathematically de®ned except for the Haar
wavelet. Most Daubechies wavelets are not symmetrical.
The Symlet family of 45 wavelets is more symmetrical
than the Daubechies but is still not exactly symmetric.
Ingrid Daubechies modi®ed her family of wavelets so that
their symmetry would be increased while they remained
simple. Only the ®rst 15 wavelets in this family were
examined because their increasing complexity requires
much more computation time. For example, on the same
computer under similar conditions, the denoising of the
same sample with a Daubechies 15 wavelet took about.27 s
and a Symlet 15 took about 1.27 s and using a Daubechies
25 wavelet it took about 0.50 s of computation time but
using a Symlet 25 wavelet it increased to 54.38 s.
The Coi¯et family of 5 wavelets was built by Daubechies
at the request of Coifman. These wavelets are more symmetrical than the Daubechies wavelets.
For each decomposition level, the signal is passed
through quadrature ®lter banks resulting in approximations
and details. This decomposition can be performed for many
levels with the decomposition process being iterated for
successive approximations. For every wavelet, decomposition levels of N from 1 to 10 were tested.
soft thresholding and hard thresholding which are used in
the matlab wavelet toolbox [9]. The de®nitions of the two
methods of thresholding are given below where x0 represents the threshold and x denotes signal [9].
6.3.2. Choosing threshold parameters
The two common methods of thresholding a signal are
Sqtwolog
Hard Thresholding : thresholded signal x for uxu
. x 0 ; and 0 for uxu # x0
Soft Thresholding : thresholding signal
sign x uxu 2 x0 for uxu . x0 ; and 0 for uxu # x0
In other words, hard thresholding is setting the elements
whose absolute values are less than the threshold to zero.
For soft thresholding, the elements whose absolute values
are lower than the threshold are set to zero, and then the
nonzero coef®cients remaining are shrunk towards zero.
Although hard thresholding is the simplest method, soft
thresholding can produce better results than hard thresholding. The reason for this is that hard thresholding may cause
discontinuities at ^x0 because those values less than the
threshold are set to zero [9].
There are four threshold selection rules that are available
to use with the wavelet toolbox [9] and are listed in Table 1.
These threshold selection rules use statistical regression of
the noisy coef®cients over time to obtain a non-parametric
estimation of the reconstructed signal without noise. For the
soft threshold estimator in the ®rst method, a threshold
selection rule which is based on Stein's unbiased estimate
of risk (SURE), is used. An estimation of risk for a certain
threshold value x0 is obtained, and then by minimizing the
risks in x0, a selection of the threshold value is obtained. The
second method uses a ®xed form threshold, which results in
minimax performance multiplied by a factor proportional to
logarithm of the length of the signal. The third method is a
combination of the ®rst and second methods. If the signalto-noise ratio (SNR) is very small (for the third method), the
SURE estimate is very noisy. If the SNR is very small and
the SURE estimate is very noisy, then the ®xed form threshold is used. The fourth method uses a ®xed threshold which
is chosen to give minimax performance for mean square
error. The minimax principle is used in the ®eld of statistics
to achieve the `minimum of the maximum mean square
error.' Fig. 6 shows the results of each of the four threshold
selection rules applied to a noisy signal.
Table 1
Threshold selection rules
Rule name
Description
Rigrsure
Selection using the principle of Stein's Unbiased
Risk Estimate (SURE)
Fixed form threshold equal to the square root of two
times the logarithm of the length of the signal
Selection using a mixture of the ®rst two options
mentioned
Threshold selection using the minimax principle
Heursure
1
A wavelet basis is orthogonal if every wavelet is at ninety degrees to
every other wavelet.
Minimaxi
S.R. Messer et al. / Microelectronics Journal 32 (2001) 931±941
937
6.6. Measuring the results of denoising
Fig. 6. (a) Original signal (b) Noisy signal Ð Signal to noise ratio 2
(c) Denoised signal Ð heuristic SURE (d) Denoised signal Ð SURE
(e) Denoised signal Ð ®xed form threshold (f) Denoised signal Ð minimax (All are denoised using a Daubechies 10 wavelet with 10 levels of
decomposition).
The basic noise model used for wavelet denoising in the
matlab wavelet toolbox is: s n f n 1 se n where s is
the complete signal, f is the signal without noise, e is the
noise, s is the strength of the noise, and time n is equally
spaced [9]. The objective of the denoising process is to
suppress the noisy part of the signal s and recover f,
which is the signal without noise.
There are three threshold rescaling methods available in
matlab wavelet toolbox which are `one', `sln', and `mln'
[9]. The scheme `one' follows the basic noise model. The
option `sln' corresponds to the basic noise model with
unscaled noise and performs threshold rescaling using
only a single estimation of level noise which is computed
based on the ®rst level coef®cients of the decomposition.
The `mln' method corresponds to the basic noise model with
non-white noise and performs threshold rescaling which is
based on a level-dependent estimation of the noise at that
decomposition level.
6.4. The use of averaging to denoise
Averaging in combination with wavelet denoising is
examined in the context of how much improvement there
is in the SNR and how many cycles should be averaged.
6.5. Use of HT
HT is used to demonstrate how effective de-noising is.
The instantaneous frequency of a signal may be used to
demonstrate how effective denoising is [14]. The use of
the instantaneous signal parameters are also explored as a
means of extracting additional information from the PCG.
SNR is a traditional method of measuring the amount
of noise present in a signal. SNR is de®ned as 10 £
log 10 Powersignal =Powernoise measured in decibels. Two
tests are performed using the SNR to measure the performance of wavelet denoising [5]. Because there is currently
no known method to calculate which wavelet and thresholding parameters best denoise a signal, tests must be
performed to evaluate the denoising capabilities of wavelets
and thresholding parameter combinations. A known amount
of noise was added to a `clean' heart sound recording.
(`Clean' refers to the fact that although attempts were
made to eliminate all environmental noise during the
recording, there is still some noise present in small
amounts.) Using various parameters, the wavelet denoising
technique was applied to the heart sound recording
which has noise added. Then the SNR was calculated for
the denoised signal and the original signal. The higher the
SNR, the less noise there is. Another test is to apply the
wavelet denoising technique to a clean recording and
compute the SNR of the resultant signal and the original
signal. This test determines the loss in the information in
the original signal by the denoising process. In other words,
the more of the original clean signal that remains after
applying the wavelet transform, the better.
The concept of adding a known amount of noise to clean
heartbeats, then denoising the signal, and seeing how much
noise remains is also employed to measure how well averaging performs as a denoising technique.
6.7. Experimental results
While no one wavelet seemed to consistently give better
results than another, there are some recommendations that
can be made. The Coi¯et 4 and 5, the Daubechies 11, 14,
and 20 and the Symlet 9, 11, and 14 seemed to produce
slightly better results in terms of noise removal than most
of the wavelets tried, while the Daubechies 1, Coi¯et 1, and
Symlet 1 and 2 seemed to produce marginally worse results.
The lower order wavelets did not perform as well as their
higher order components due to their properties such as
support length, regularity, and the number of vanishing
moments [18].
It appeared that using 5 levels of decomposition usually
proved adequate. For three trials (a trial being a clean heartbeat), varying amounts of noise was added (1, 5, and
10 dBs) and the levels of decomposition were varied from
1 to 10 with other denoising parameters remaining constant.
In most cases, using a decomposition level of 5 achieved an
SNR after denoising within 95% of the best denoising result
for that set of data. However, in a test performed where
noise was added to three trials in increments of 1 dB from
1 to 20 dBs, the best denoising results obtained in each case
ranged from using 4 to 10 levels of decomposition.
The wavelet transform process of decomposition and
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S.R. Messer et al. / Microelectronics Journal 32 (2001) 931±941
Fig. 7. This ®gure demonstrates that hard thresholding can cause discontinuities in a signal. (a) characteristic heartbeat signal (b) the heartbeat with
white noise added resulting in an SNR of 1 dB (c) the heartbeat denoised
using soft thresholding (d) the heartbeat denoised with hard thresholding.
recomposition was applied to several clean heartbeats. The
SNR of the original signal and the signal after WT is applied
are calculated. This number represents the loss in information in the wavelet analysis process. Any wavelet that
consistently gave an SNR of under 60 dBs was considered
to have lost too much information from the original signal.
The wavelets which seemed to lose the most information
were the Daubechies 1, 2, 3, 43, 44, 45 and the Symlet 1
and 2.
Soft thresholding de®nitely outperformed hard thresholding in the threshold selection category. Soft thresholding
almost always gave better SNR after denoising than hard
thresholding. Fig. 7 shows how hard thresholding may cause
discontinuities in a signal.
Of the four threshold selection rules, the minimax and
Fig. 8. This ®gure is a comparison of the different threshold rescaling
methods. Noise in increments of 1 dB from 1 to 20 dBs is added to a
clean characteristic heartbeat. The signal is then denoised using wavelet
denoising (Daubechies 14 wavelet with 10 levels of decomposition using
soft thresholding, and the rigorous SURE threshold selection rule) varying
the threshold rescaling methods. Overall, the `sln' method performs the
best.
Fig. 9. Best SNR results before denoising versus after wavelet denoising for
three trials. Each trial is a different, clean heartbeat. A known amount of
white noise in decibels is added to a characteristic heartbeat. Different
combinations of wavelets, thresholding techniques, and levels of decomposition are tried. The highest SNR after denoising is displayed for each
trial. It is interesting to note that the best results for each trial are very
similar.
SURE (`rigrsure') threshold selection schemes are more
conservative than the others, and therefore should be used
when small details of signal lie in the noise range. The two
other schemes remove the noise more aggressively. Because
small details of the PCG signal are located in the noise
range, it was believed that the `rigrsure' method would be
more effective. Our expectations proved to be true. The
Fig. 10. This chart shows the SNR after adding white noise to a series of
heartbeats and the SNR after averaging these series of heartbeats to obtain a
characteristic heartbeat and reduce noise. There seems to be marginal
improvement in SNR when little noise is present in the signal and the signal
is not averaged a fair number of times. For example, with an SNR of 1 dB,
after averaging the signal 10 times, we see the noise levels decrease as the
SNR approaches 11 dBs, but with an initial SNR of 20 dBs and averaging
10 heartbeats, the SNR is still about 20 dBs meaning the noise remains.
Averaging a series of 50±75 beats seems to give the best results here in
terms of recording and computation time trade-off.
S.R. Messer et al. / Microelectronics Journal 32 (2001) 931±941
Fig. 11. This graph shows the SNR after adding white noise to a series of
heartbeats versus the SNR after denoising the signal. Various methods are
tried: wavelet denoising only, averaging only, and wavelet denoising
combined with averaging. The wavelet denoising combined with averaging
is the most successful denoising method. Heartbeat signals of lengths of 30,
60 and 90 were averaged. It appears that averaging at least 60 heartbeats
would be recommended. There is a signi®cant improvement of averaging
60 beats over 30 heartbeats while averaging 90 beats over 60 beats does not
show nearly the same improvement.
rigorous SURE method almost always produced better
results than the other methods with the heuristic SURE
coming in second. Table 2 shows typical results for each
of the four threshold selection rules.
939
The `sln' method performed the best of three threshold
rescaling methods available. For example, in Fig. 8, with
large amounts of noise present, the methods all perform
roughly equally with the `sln' method producing slightly
better results than the others. However, as less noise is
present, the `sln' method appears to be, by far, the best.
The `mln' method is a close second. The `one' method
gives about the same SNR after denoising no matter what
amount of noise was present in the signal.
Fig. 9 shows the best results for adding known amounts of
noise to three different heartbeats and then applying wavelet
denoising with varying parameters. The best results are very
similar proving that the results of wavelet denoising are
easily reproducible for heartbeat sounds. Also, the SNRs
after wavelet denoising appear to be relatively linear corresponding to the linear SNR between the signal and the addition of white noise.
Averaging seemed to produce signi®cant improvements
especially if there is a large amount of noise present in the
signal. Fig. 10 shows that averaging a series of 50±75 beats
seems to give the best result in terms of recording and computation time tradeoff. It is dif®cult to obtain a long, clean
recording which also increases the computation time required.
Fig. 11 shows a comparison of using wavelet denoising
alone and wavelet denoising combined with averaging. It
clearly demonstrates that combining the techniques is
much more effective. It shows that given a choice between
averaging 30, 60 or 90 beats that 60 beats provides a good
compromise in terms of denoising and recording and
computation time.
Fig. 12. This ®gure shows (a) HT of a characteristic heartbeat with 20 dBs of additive noise (b) the phase space diagram of the same noisy signal (c) the HT of
the denoised heartbeat and (d) the phase space diagram of the denoised heartbeat.
940
S.R. Messer et al. / Microelectronics Journal 32 (2001) 931±941
Table 2
Typical SNR results after denoising using four threshold selection rules.
White noise was added to three different characteristic heartbeats resulting
in an SNR of 1 dB. Wavelet denoising was applied to the heartbeats using
the same parameters (Symlet 14 wavelet with 10 levels of decomposition,
soft thresholding, and used the basic noise model with a single estimation
level of noise) except for varied threshold selection rules. The SNR was
then calculated in dBs. The `rigrsure' outperforms all other methods in
every case
Threshold
selection rule
Trial 1
Trial 2
Trial 3
`Rigrsure'
`Heursure'
`Minimaxi'
`Sqtwolog'
13.65
13.27
9.30
7.26
13.78
13.24
8.09
8.09
13.70
12.72
9.84
7.37
The HT may be used to demonstrate how effective the
denoising process performs but the instantaneous frequency
and phase space diagram are better suited for this task. It
may be seen that the signal is much cleaner after being
denoised in Fig. 4, but this dramatic improvement is only
because the signal has much additive noise. The HT of a
signal does not accentuate the noise in a signal and thus does
not show such dramatic improvement after denoising as is
seen in phase space diagrams shown in [5]. The phase space
diagrams take the derivative of a signal thus accentuating
the noisy, high frequency, content. This fact is demonstrated
in Fig. 12.
The instantaneous amplitude is an alternative method
of looking at the PCG data. Fig. 13 demonstrates that
recording the PCG of a patient is reproducible because
plots of the instantaneous amplitude of a PCG recorded
on 4 different occasions are very similar. Fig. 14 shows
the instantaneous amplitude of PCGs for patients with
various pathological conditions and patients with normal
hearts. We were limited by the number of PCG recordings available, but by examining this plot we may see
that the healthy patients appear to have a well de®ned
Fig. 14. (a) Instantaneous amplitude of a heartbeat of a patient with mitral
valve prosthesis. (b) Instantaneous amplitude of a heartbeat of a patient
with a heart murmur and hypertension. (c) Instantaneous amplitude of a
heartbeat of a patient with an aortic stenosis and hypertension. (d) Instantaneous amplitude of a heartbeat of a patient with a systolic murmur and
angioplasty. (e) Instantaneous amplitude of a heartbeat of a patient with
atrial ®brillation. (f) Instantaneous amplitude of a heartbeat of a patient with
hypertension. (g) Instantaneous amplitude of a heartbeat of a healthy
patient. (h) Instantaneous amplitude of a heartbeat of a different healthy
patient.
and compact S1 and S2 whereas some of the patients
with pathological conditions do not.
We have demonstrated that the HT has many interesting
applications to PCGs. The complex PCG trace reveals much
information about the signal. The HT may also be used to
calculate the instantaneous parameters of a signal. The
phase space diagram demonstrates denoising better visually,
instead of plotting the HT and the signal, although instantaneous frequency calculated using the HT effectively demonstrates denoising.
7. Conclusions
Fig. 13. This ®gure shows the instantaneous amplitude of 4 heartbeats
recorded at different times from the same healthy patient. They are all fairly
similar demonstrating that this technique is reproducible.
We have demonstrated that wavelet denoising techniques
and averaging are useful for removing white noise from
heart sounds. Wavelet denoising in combination with averaging produced the best denoising results of the methods
applied. However, there may be certain clinical cases, for
example if a pathological condition was only present in
some beats and not others, where wavelet denoising alone
should be employed as averaging reduces a sequence of
heartbeats to a single characteristic heartbeat.
Although there was no evidence that a single wavelet was
the best suited for denoising heartbeats, there were some
wavelets which were slightly better than others and certain
wavelets which would not be recommended for this
purpose. We reached the conclusion that a decomposition
level of 5 produced reasonable results while decomposing
S.R. Messer et al. / Microelectronics Journal 32 (2001) 931±941
the signal further often produced marginal bene®ts and
increases computation time. Soft threshold de®nitively
outperformed hard thresholding. Of the four threshold selection rules, the `rigrsure' rule performed the best, and the best
choice of the threshold rescaling methods proved to be the
`sln' method. Averaging heartbeats to produce a characteristic heartbeat signi®cantly reduced noise. Averaging a
sequence of about 60 heartbeats gave optimal denoising
bene®ts in terms of computation time and dif®culty in
recording a clean PCG.
The use of the HT was explored in relation to the analysis
of heartbeats. More information is displayed in the complex
PCG trace. The instantaneous frequency of a PCG, as a tool
of clinical use, which perhaps would reveal information
about the condition of the heart, merits further research.
Another method of denoising the PCG which should be
explored in the future would be the matching pursuit method
[19]. Also, more research is needed to determine what sort
of noise corrupts PCGs. Once, the sort of noise polluting the
PCG is determined, a system could employ different denoising techniques based upon what sort of noise was present.
Acknowledgements
We would like to thank Leonard Hall, Jarrad Maple and
Mohammed Ali Tinati (all of the Department of Electrical
and Electronic Engineering, The University of Adelaide) for
allowing us to make use of their heart sound recording
equipment and software.
We would also like to thank colleagues at the Centre
for Biomedical Engineering (Electrical and Electronic
Engineering, The University of Adelaide) including Brad
Ferguson, Leonard Hall, Greg Harmer, and Samuel Mickan,
for providing assistance and inputs in various areas along
the way.
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